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Mirrors > Home > MPE Home > Th. List > volf | Structured version Visualization version GIF version |
Description: The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.) |
Ref | Expression |
---|---|
volf | ⊢ vol:dom vol⟶(0[,]+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovolf 24085 | . . . . . 6 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞) | |
2 | ffun 6519 | . . . . . 6 ⊢ (vol*:𝒫 ℝ⟶(0[,]+∞) → Fun vol*) | |
3 | funres 6399 | . . . . . 6 ⊢ (Fun vol* → Fun (vol* ↾ dom vol)) | |
4 | 1, 2, 3 | mp2b 10 | . . . . 5 ⊢ Fun (vol* ↾ dom vol) |
5 | volres 24131 | . . . . . 6 ⊢ vol = (vol* ↾ dom vol) | |
6 | 5 | funeqi 6378 | . . . . 5 ⊢ (Fun vol ↔ Fun (vol* ↾ dom vol)) |
7 | 4, 6 | mpbir 233 | . . . 4 ⊢ Fun vol |
8 | resss 5880 | . . . . . 6 ⊢ (vol* ↾ dom vol) ⊆ vol* | |
9 | 5, 8 | eqsstri 4003 | . . . . 5 ⊢ vol ⊆ vol* |
10 | fssxp 6536 | . . . . . 6 ⊢ (vol*:𝒫 ℝ⟶(0[,]+∞) → vol* ⊆ (𝒫 ℝ × (0[,]+∞))) | |
11 | 1, 10 | ax-mp 5 | . . . . 5 ⊢ vol* ⊆ (𝒫 ℝ × (0[,]+∞)) |
12 | 9, 11 | sstri 3978 | . . . 4 ⊢ vol ⊆ (𝒫 ℝ × (0[,]+∞)) |
13 | 7, 12 | pm3.2i 473 | . . 3 ⊢ (Fun vol ∧ vol ⊆ (𝒫 ℝ × (0[,]+∞))) |
14 | funssxp 6537 | . . 3 ⊢ ((Fun vol ∧ vol ⊆ (𝒫 ℝ × (0[,]+∞))) ↔ (vol:dom vol⟶(0[,]+∞) ∧ dom vol ⊆ 𝒫 ℝ)) | |
15 | 13, 14 | mpbi 232 | . 2 ⊢ (vol:dom vol⟶(0[,]+∞) ∧ dom vol ⊆ 𝒫 ℝ) |
16 | 15 | simpli 486 | 1 ⊢ vol:dom vol⟶(0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ⊆ wss 3938 𝒫 cpw 4541 × cxp 5555 dom cdm 5557 ↾ cres 5559 Fun wfun 6351 ⟶wf 6353 (class class class)co 7158 ℝcr 10538 0cc0 10539 +∞cpnf 10674 [,]cicc 12744 vol*covol 24065 volcvol 24066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-icc 12748 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-ovol 24067 df-vol 24068 |
This theorem is referenced by: volsup 24159 volsup2 24208 volivth 24210 itg1climres 24317 itg2const2 24344 itg2gt0 24363 areambl 25538 voliune 31490 volfiniune 31491 volmeas 31492 volsupnfl 34939 areacirc 34989 arearect 39829 areaquad 39830 volioof 42279 volicoff 42287 voliooicof 42288 fourierdlem87 42485 voliunsge0lem 42761 volmea 42763 hoidmv1lelem1 42880 hoidmv1lelem2 42881 hoidmv1lelem3 42882 ovolval4lem1 42938 ovolval5lem1 42941 |
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